Abstract
Publisher Summary This chapter presents the interplay between the symmetries of a partial differential equation, the coordinate systems in which the equation admits solutions via separation of variables, and the properties of the separated solutions. The chapter discusses the wave equation in three-dimensional space time. The chapter illustrates the intimate connection between the theory of linear and Hamilton–Jacobi differential equations in terms of symmetry and separation of variables. It also discusses the subspace S' k of homogeneous symmetric kth-order elements in the enveloping algebra with the space K k of kth-order polynomials in the basic functions. It further reviews the well-known relationship between a first-order partial differential equation and the Hamiltonian system of ordinary differential equations.
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